In very broad strokes, you need several things:
A server-side database to store user information. You’re not going to create an actual user account on the server. You’re going to create a database of website users, and when a person logs in, you’ll compare their user/pass to the ones stored in the db.
Since you’re asking the question, it’s fair to assume you have no experience with the different schemes in #1 above. Do you have a server you want to use or do you have resources available that would weight that choice? If you’re not a programmer and don’t want to use a canned system like vBB, you need to pick a language and start learning. Holy wars have been fought over which language is better, but the distinctions are really pretty minor and all of the ones mentioned above (and others) will do what you want. There are also a lot of generic systems for handling various functionality which wouldn’t require you to start from scratch.